A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start.
How many Hamiltonian circuits can you find in these graphs?

More on Mazes

Stage: 2 and 3

Article by NRICH team

Published February 2000,February 2011.

In February 1999, we had the story of
an ancient Celtic maze . There is a long tradition of creating
mazes throughout history and across the world. Some mazes told
stories, some had religious meaning, some were for defence against
invaders and some were made as outdoor amusements for rich people.
There are now many public mazes where people can enjoy the
challenge of trying to find their way through the maze.

If you like looking at mazes, there's an excellent English
website (http://www.mazemaker.com/) that
has pictures of many different types of mazes that can be visited.
For example; the hedge mazes at Leeds Castle or Edinburgh Zoo; or
the mirror maze at Wookey Hole Caves; or the stone path maze in
Bath. The man who created this website is a maze designer and so
has collected a huge range of mazes. There are databases of 'Mazes
of the World', cornfield mazes and 'Mazes in Outer Space'. If you'd
like to try doing a maze puzzle, try 'Queen's Maze', or 'Tumbling
Log Maze (As always - never make contact with anyone over the
internet without asking your parents).

The most common mazes we see are those drawn on paper as puzzles.
You can eventually solve a maze by just trying all the paths until
you get lucky, but you think about what you are doing you can
become quite clever at finding the way quickly. Some mazes are
drawn very carefully using a traditional method or a mathematical
pattern. If you'd like to find out more about this, visit the
Through Mazes
to Mathematics website.

Lewis Carroll (famous for writing Alice in Wonderland) was a
mathematician. When he was in his twenties he drew a very tricky
maze with paths that crossed under and over each other. The Year 7
Maths Club from Bexley Grammar School suggested this maze a
challenge. If you click here on Lewis Carroll's Maze , you will go
to a page that can be printed off. Challenge your friends to find a
way from the middle to the outside before you do!

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.